The Hodge diamond of O’Grady’s six-dimensional example

Abstract

We realize O'Grady's six dimensional example of irreducible holomorphic symplectic manifold as a quotient of an IHS manifold of K3 [3] -type by a birational involution, thereby computing its Hodge numbers. IntroductionIn this paper we present a new way of obtaining O'Grady's six dimensional example of irreducible holomorphic symplectic manifold and use this to compute its Hodge numbers. Further applications, such as the description of the movable cone or the answer to Torelli-type questions for this deformation class of irreducible holomorphic symplectic manifolds, will be the topic of a subsequent paper. Recall that an irreducible holomorphic symplectic manifold (IHS) is a simply connected compact Kähler manifold that has a unique up to scalar holomorphic symplectic form. They arise naturally as one of the three building blocks of manifolds with trivial first Chern class according to the Beauville-Bogomolov decomposition [6], [2], the other two blocks being Abelian varieties and Calabi-Yau manifolds. By definition, IHS manifolds are higher dimensional generalizations of K3 surfaces, moreover they have a canonically defined quadratic form on their integral second cohomology group, which allows to speak of their periods and to develop their theory in a way which is analogous to the theory of K3 surfaces. The interested reader can see [16] and [39] for a general introduction on the topic. There are two deformation classes of IHS manifolds in every even dimension greater or equal to 4, introduced by Beauville in [2]. They are the Hilbert scheme of n points on a K3 and the generalized Kummer variety of dimension 2n of an abelian surface (i.e. the Albanese fiber of the Hilbert scheme of n+1 points of the abelian surface). Elements of these two deformation classes have second Betti number equal to 23 and 7, respectively, and are referred to as IHS manifolds of K3 [n] -type and of generalized Kummer type, respectively. There are two more examples, found by O'Grady in [37] and [38], of dimension ten and six, respectively, which are obtained from a symplectic resolution of some singular moduli spaces of sheaves on a K3 surface and on an abelian surface, respectively. They are referred to as the exceptional examples of IHS, and their deformation classes are denoted by OG10, respectively OG6. These exceptional examples have not been studied as much and their geometries are less understood. Though their topological Euler characteristic is known, see [42] and [31], even other basic invariants such as their Hodge numbers have not been computed yet. In the case of manifolds of K3 [n] -type, the Hodge numbers were computed by Göttsche [11]. One of the main results of this paper is to realize O'Grady's six dimensional example as a quotient of an IHS manifold of K3 [3] -type by a birational symplectic involution: we therefore relate this deformation class to the most studied deformation class of IHS manifolds and this allows us, by resolving the indeterminacy locus of the involution and by describing explicitly its fixed locus (wh...

Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds

“…In dimension 6, we have 3 topological models, namely S [3] , K 3 (A) and OG6 constructed in [20], and their classes are linearly independent, as proves the following computation. The Chern numbers c 3 2 , c 2 c 4 , c 6 of K3 [3] are computed in [3], those of K 3 (A) are computed in [17], and those of OG6 are computed in [14]. Thanks to these works, the matrix of Chern numbers for these three varieties takes the form (where the first line indicates the Chern numbers of K3 [3] , the second line those of K 3 (A), and the third line those of OG6):   36800 14720 3200 30208 6784 448 30720 7680 1920   .…”

Topological bounds on hyperkähler manifolds

“…In dimensions six and eight we have Hilbert schemes of points on K3 surfaces and generalized Kummer varieties; their Rozansky-Witten invariants were calculated in Sawon [19], and one can readily verify the conjecture is true for these examples. In dimension six there is one additional example due to O'Grady [15], whose Hodge numbers were calculated by Mongardi, Rapagnetta, and Saccà [14]. In six dimensions the Rozansky-Witten invariants can be computed in terms of the Hodge numbers, and this leads to a verification of the conjecture for O'Grady's example too.…”

Hodge Numbers of O'Grady 6 Via Ngô Strings

“…The Betti and Hodge numbers of manifolds of OG6 type were first computed by Mongardi, Rapagnetta, and Saccà in [MRS18]. The purpose of this paper is to provide a new computation of the Betti and Hodge numbers of these manifolds using the method of Ngô strings introduced by de Cataldo, Rapagnetta, and Saccà in [dCRS19].…”